Group #20274206 — subgroups

Subgroups — #20274206

Order 6784.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810905	C₂	1	2	yes	maximal
2	#16810753	C₂	1	1	yes	maximal, characteristic
3	#16810374	C₂	1	1	yes	maximal, characteristic
4	#16810369	C₂	1	2	yes	maximal
5	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	#13133148	—	2	2	no	—
8	C₄⋊#88322	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133146	—	2	1	no	—
11	#13133140	—	2	2	no	—
12	#13133127	—	2	1	no	—
13	C₂×Q₁₆×C₅₃	—	2	1	no	—
14	C₂×Q₁₆×C₅₃	—	2	1	no	—
15	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
19	Q₁₆×C₅₃	—	4	2	no	—
20	Q₁₆×C₅₃	—	4	2	no	—
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
27	#11719919	—	2	1	no	—
28	C₄×Dic₅₃	—	2	1	no	—
29	C₂×#88322	—	2	1	no	—
30	C₂×#88322	—	2	2	no	—
31	Q₈×C₅₃	—	4	2	no	—
32	Q₈×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
37	C₂×Dic₅₃	—	2	1	no	—
38	C₂×Dic₅₃	—	2	1	no	—
39	Dic₁₀₆	—	2	2	no	—
40	C₈×C₅₃	—	2	1	no	—
41	C₈×C₅₃	SD₁₆	1	2	yes	—
42	#88322	—	4	1	no	—
43	#88322	—	4	2	no	—
44	C₂²×C₅₃	#121	1	1	yes	characteristic
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	8	2	no	—
47	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
48	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₄×C₅₃	#137	1	2	yes	—
50	Dic₅₃	—	4	1	no	—
51	Dic₅₃	—	4	1	no	—
52	#1690	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	#460	1	1	yes	characteristic
54	C₂×C₅₃	#470	1	1	yes	characteristic
55	C₂×C₅₃	#473	1	1	yes	characteristic
56	#494	D₅₃	1	1	yes	characteristic
57	#475	—	53	2	no	—
58	#446	—	53	1	no	—
59	#334	—	53	1	no	—
60	#332	—	53	2	no	—
61	C₅₃	#1690	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	2	1	no	—
63	C₂×Q₁₆	—	2	1	no	—
64	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
65	C₄×Q₈	—	53	1	no	—
66	#106	—	106	2	no	—
67	C₄⋊C₈	—	53	1	no	—
68	C₄⋊C₈	—	53	2	no	—
69	C₄⋊C₈	—	106	1	no	—
70	#103	—	106	2	no	—
71	#97	—	106	1	no	—
72	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	53	1	no	—
75	Q₁₆	—	4	2	no	—
76	Q₁₆	—	4	2	no	—
77	C₂×C₈	—	106	2	no	—
78	C₂×C₈	—	106	1	no	—
79	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	53	1	no	—
84	C₄⋊C₄	—	106	1	no	—
85	C₄²	—	106	1	no	—
86	C₄²	C₂²×D₅₃	1	1	yes	characteristic
87	Q₈	—	4	2	no	—
88	Q₈	—	4	2	no	—
89	Q₈	—	106	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	106	1	no	—
92	C₂×C₄	—	106	1	no	—
93	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
94	C₂×C₄	#11719947	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	212	1	no	—
98	C₈	—	212	2	no	—
99	C₈	#11719923	1	2	yes	—
100	C₂²	#13133188	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	8	2	no	—
103	C₄	—	212	1	no	—
104	C₄	—	212	1	no	—
105	C₄	C₂×#11719923	1	1	yes	characteristic
106	C₄	#13133243	1	1	yes	characteristic
107	C₄	#13133249	1	2	yes	—
108	C₂	#16810820	1	1	yes	characteristic
109	C₂	#16810878	1	1	yes	characteristic
110	C₂	#16810895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810905

C₂

yes

maximal

#16810753

C₂

yes

maximal, characteristic

#16810374

C₂

yes

maximal, characteristic

#16810369

C₂

yes

maximal

C₅₃×#494

C₂

yes

maximal, characteristic

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133148

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133146

—

#13133140

—

#13133127

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

—

#88322

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

C₅₃

#1690

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₅₃

yes

—

C₄×Q₈

—

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

—

106

—

#97

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

106

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719947

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₈

—

212

—

C₈

#11719923

yes

—

100

C₂²

#13133188

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

212

—

104

C₄

—

212

—

105

C₄

C₂×#11719923

yes

characteristic

106

C₄

#13133243

yes

characteristic

107

C₄

#13133249

yes

—

108

C₂

#16810820

yes

characteristic

109

C₂

#16810878

yes

characteristic

110

C₂

#16810895

yes

characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	#121
hypercenter	#494	D₅₃
hypocenter	C₅₃	#1690
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#13133188

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#494

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
61. C₅₃	#1690	semidirect

H ≅ G/N

type

61. C₅₃

#1690

semidirect

Subgroups — #20274206

Conjugacy classes of subgroups

Complements